To solve this problem I first thought on what numbers would work in the boxes and if the other numbers work with it. I started with closest to 5000 and thought of 4 numbers to go in the box and then did the same with the rest. Once I had gone through all of the numbers I worked out what numbers go together ,I had solved the problem quickly.
I also thought if I left at-least 1 number which would work in a solution this might work. On some of the numbers this worked but on some it didn't.
The largest even number was easy to work out because you would only need either 2 4 6 8 or 0 for the units column, this is the same with the largest odd number but with 1 3 5 7 or 9. the rest then just slotted into place.
MY ANSWERS:
LARGEST ODD NUMBER: 9443
LARGEST EVEN NUMBER: 9636
LARGEST MULTIPLE OF 3: 7788
SMALLEST MULTIPLE OF 5: 1225
CLOSEST TO 5000: 5001
All of this works.